Operators and observables
PHYS 410 · Quantum Formalism
Quantum observables are represented by operators acting on states. This lesson explains linear operators, Hermitian operators, commutators, compatible observables, and measurement.
Key equations
hat{A}|psi
angle=|phi
anglehat{A}(a|psi
angle+b|phi
angle)=ahat{A}|psi
angle+bhat{A}|phi
anglehat{x}psi(x)=xpsi(x)hat{p}=-ihbar rac{d}{dx}langle phi|hat{A}psi
angle=langle hat{A}phi|psi
anglehat{A}|a
angle=a|a
anglehat{H}|E_n
angle=E_n|E_n
angle|psi
angle=sum_n c_n|a_n
angleP(a_n)=|c_n|^2[hat{A},hat{B}]=hat{A}hat{B}-hat{B}hat{A}[hat{A},hat{B}]=0[hat{x},hat{p}]=ihbarLearning objectives
- Define linear operators on quantum states.
- Identify position and momentum operators in position space.
- Explain why observables are Hermitian.
- Interpret eigenvalue equations in measurement.
- Use commutators to discuss compatible observables.
Operators act on states
In quantum mechanics, physical observables such as position, momentum, energy, and angular momentum are represented by operators. An operator acts on a state vector to produce another vector:
angle=|phi angle$$ Operators are usually linear. Linearity means $$hat{A}(a|psi angle+b|phi angle)=ahat{A}|psi angle+bhat{A}|phi angle$$ This preserves the superposition structure of quantum theory. ## Position and momentum operators In the position representation, the position operator acts by multiplication: $$hat{x}psi(x)=xpsi(x)$$ The momentum operator acts as a derivative: $$hat{p}=-ihbar rac{d}{dx}$$ These operators do not commute, which is the mathematical origin of position-momentum uncertainty. ## Hermitian operators Observable quantities must have real measurement results. Therefore observables are represented by Hermitian operators. An operator $hat{A}$ is Hermitian if $$langle phi|hat{A}psi angle=langle hat{A}phi|psi angle$$ for suitable states. Hermitian operators have real eigenvalues and orthogonal eigenvectors for distinct eigenvalues. ## Eigenvalue equations If a state $|a angle$ satisfies $$hat{A}|a angle=a|a angle$$ then $|a angle$ is an eigenstate of $hat{A}$ with eigenvalue $a$. If the system is in that eigenstate, measuring $A$ gives the value $a$ with certainty. For example, energy eigenstates satisfy $$hat{H}|E_n angle=E_n|E_n angle$$ ## Measurement and expansion If a state is expanded in eigenstates of an observable, $$|psi angle=sum_n c_n|a_n angle$$ then measuring $A$ gives result $a_n$ with probability $$P(a_n)=|c_n|^2$$ assuming normalized states. After an ideal measurement yielding $a_n$, the state is projected into the corresponding eigenstate or eigenspace. ## Commutators The commutator of two operators is $$[hat{A},hat{B}]=hat{A}hat{B}-hat{B}hat{A}$$ If $$[hat{A},hat{B}]=0$$ then the observables are compatible in the sense that they can often have simultaneous eigenstates. If the commutator is nonzero, simultaneous definite values are generally impossible. The canonical example is $$[hat{x},hat{p}]=ihbar$$ ## Complete sets of commuting observables In more complex systems, a single observable may not fully specify a state because of degeneracy. A complete set of commuting observables provides enough compatible measurements to label states uniquely. For hydrogen, energy, angular momentum magnitude, and one angular momentum component are often used together. ## The big idea Operators represent physical observables in quantum mechanics. Hermitian operators guarantee real measurement results, eigenstates represent states with definite values, and commutators encode compatibility or uncertainty. Measurement probabilities come from expanding states in the eigenbasis of the observable being measured.Ask your AI physics guide
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